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suaded himself that he had experienced in his own person, a fact alluded to by Horace (Lib. I. Ode 28.)、


Tartara Panthoïden, iterum Orco.

Demissum, quamvis clypeo Trojana refixo
Tempora testatus, nihil ultra

Nervos atque cutem morti concesserat atræ.'

We have referred to this characteristic trait of Pythagoras by way of illustrating the compatibility of exact studies with the exercise even of the wildest imagination. We can barely allude to a series of authors, all of whom were distinguished in their times, and some of whom left behind them memorials of their genius, which have lived to our day,-Enopides, who lived nearly five centuries before Christ,-Zenodorus, a contemporary of his, who showed, that figures with equal circumferences have not therefore necessarily equal areas,-Hippocrates, who made the first discovery of certain curvilinear. spaces being equal to certain rectilinear, and distinguished himself by his attempts to solve the problem of the duplication of the cube, one of the most celebrated propositions in ancient geometry, said to have originated in a response of the Delian oracle, directing the Athenians to appease the wrath of Apollo, by doubling the altar in his temple, which happened to be a perfect cube. Passing over these names and others without further comment, we come to Plato, who, for forty years, presided over the Academy with unrivalled genius and eloquence. The characters of such men as Plato, we contemplate with mingled pride and reverence; with pride, for they show the high powers with which our nature is gifted, and the universality of true genius; and with reverence, for they have raised themselves, by the assiduous cultivation of their immortal intellects, to a dignity of character which we view as one of almost inaccessible glory. This wonderful man possessed by nature a spirit keenly alive to lofty and ennobling sentiments; he possessed by education a taste which realized the excellence of the beautiful and the sublime. Poetry and music were the elegant amusements by which he not only unbent but refined his mind. In his earlier days he is said to have written many metrical pieces, which, when he gave himself to the pursuit of philosophy, under the instruction of Socrates, he destroyed; but in his extant works we yet discover a sufficient vein of that noble enthusiasm, to convince us of the inexhaustible riches of the original mine. Gifted with these lofty

powers and ennobling sentiments, which, according to the popular impression, do not chime well with a love of mathematics, Plato not only was a perfect master of geometry himself, as far as the science had been cultivated, and made many additions by his own discoveries, but insisted that it should be made a leading object of study by his disciples. Every schoolboy knows the inscription over his door, Let no one enter here who is ignorant of geometry.' It is not wonderful that the influence of Plato should have given au impulse to the science, which lasted long and contributed much to its advancement. An illustrious school of geometricians formed under his auspices, embracing the names of Aristæus, Eudoxus, Menæchmus, and many others, gave themselves to the pursuit with an enthusiasm worthy of the cause, and wrought improvements corresponding to their talents and zeal. One of the first writers, who collected the scattered elements of geometry into a regular system, was Euclid, an author of the Alexandrian school. The fame which his treatise has ever enjoyed, is one of the most striking instances of lasting literary glory, that the history of man has to offer. Down to the present day it has been used, almost exclusively, in the mathematical schools of the civilized world, while Aristotle, who, for many centuries, exerted a most despotic power over the reason of men, has been gradually banished from the halls of learning.

Archimedes of Syracuse flourished about half a century. later than Euclid. His name, from its having been recorded in Roman history, is more familiar to general readers than that of any other ancient mathematician. As a geometer, he undoubtedly has a claim to be placed first in the first rank. The enthusiasm with which he regarded the cultivation of his favorite science, and his disinterested and almost passionate devotion to the cause of truth, without reference to its application to purposes of utility, may be regarded as a striking instance of the moral sublime. His method of approximating the ratios of incommensurables, has served as a guide to all succeeding geometers. We know of no fairer title to be placed among the great lights of mankind, than is given by such numerous and important discoveries as were made by Archimedes; and the noble consciousness of his greatness, and his faith in the just appreciation of posterity, place him on the same roll with Galileo and Bacon. Few things, recorded in ancient history, are oftener alluded to, than the destructive effects

which he was enabled to produce at the siege of Syracuse, by means of his knowledge of geometry and mechanics. Science then achieved one of her sublimest triumphs over brute force and the ordinary apparatus of warfare.

'Against these naval forces,' says Livy, Archimedes distributed along the walls engines of various magnitudes. The ships which lay at a distance, he assailed with rocks of great weight; those which were nearer the city, he attacked with lighter and more numerous weapons; and at last, that his countrymen might be protected against the enemy, while discharging upon him their weapons, he made in the wall, from top to bottom, numerous apertures, a cubit in diameter, through which, under cover of the wall, part of them harassed the enemy with arrows, and part with darts shot from crossbows. Those ships which approached nearer, in order that the engines might overshoot them, were annoyed by a long lever (tolleno), one arm of which projected from the wall. An iron grapple was secured to its extremity by a strong chain, and this grapple fastening upon the prow of the ship, raised it upon the stern by means of a counterpoise of lead, which brought to the ground the other arm of the lever; then being suddenly disengaged, the ship fell from the wall, and was dashed into the water, to the terror of the sailors, with such violence, that, even if it had fallen in an upright position, it would have dipped much water. In this way the attack by sea was completely frustrated; and all the forces of the enemy were concentrated in preparation for an attack by land. But on this side also, the city was defended by a similar apparatus of engines, prepared at the expense of Hiero, during a course of many years, by the unrivalled ingenuity of Archimedes.' *

We have quoted this passage at length, not only because it refers to one of the most remarkable military operations of the Romans, but because it is the most remarkable instance of a military defence being protracted by the aids of science, that we find in the war annals of antiquity. Archimedes considered himself as descending from the dignity of the pursuit of abstract truth, to turn the powers of knowledge even against the enemies of his country. He wished that his glory should rest upon the permanent basis of the discovery of geometrical truth;

*Lib. xxiv. c. 34.

and in this his anticipations and hopes have been realized. That his name might be perpetually connected with the memory of his discoveries, he directed that a sphere, inscribed in a cylinder, should be engraved on his tomb, thus making his most brilliant intellectual exploit his only, and, we may add, his most glorious epitaph. It is impossible to contemplate the character of this great man, without feeling that he experienced the same poetical and lofty aspirations after fame, which have always been, we believe, the accompaniments of greatness. The prophetic foresight of Horace, the passionate visions of Cicero, the glowing but solemn confidence of Bacon, justify the spirit of a French philosopher's remark, that, in loftiness of intellectual character, Homer and Archimedes stand upon the same level. But we must break away from these reflections, and enter upon the examination of the work, whose title we have placed at the head of the present article.

Mr Walker is well known as a successful teacher of mathematics, in the celebrated school of Messrs Cogswell and Bancroft, in Northampton. Experience is the only safe test of the merit of an elementary work, in any department of knowledge, designed for the instruction of beginners; and the book before us contains the elements of geometry, moulded to that form, which some years of practical acquaintance with the art of teaching, suggested as the best. The best modern treatise on geometry, compiled from ancient and modern authors, and uniting the excellences of all to an extraordinary degree, is undoubtedly that of Legendre. But on many

accounts this is unfit for the use of schools. As an analysis of the science we hold it above all praise. But Legendre, though he departed, in some respects, from the rigid methods of the ancients, did not depart enough from them to avoid a degree of prolixity, which renders his treatise too cumbrous and expensive for a manual in common schools. Without in the least disparaging the merit of that eminent and judicious mathematician, we may assert, and we believe our assertion will be borne out by universal experience, that his work has not supplied the want of an elementary treatise of geometry, for the ordinary and general purposes of instruction; which want, Mr Walker has attempted to supply.

Legendre's work is divided into eight sections, four of which treat of plane geometry, and four of solid geometry. This

division is well enough, but appears a little arbitrary. A somewhat different, and, as we think, a more perspicuous arrangement for beginners, has been adopted by Mr Walker. On this point, let the author speak for himself.

The division of the work into three sections is founded in the nature of the subject. Extension, or the space which matter occupies, has three dimensions, length, breadth, and thickness. These may be considered separately or in connexion. When we consider length alone, its representative is a line. Hence the first section treats of lines and their relations. When we consider length and breadth together, or length in two ways, their representative is a surface. Hence the second section treats of surfaces. Lastly, when we consider length, breadth, and thickness together, or length in three ways, their representative is a solid. Hence the third section treats of solids.'

This arrangement is clear, and the reasons for it are strong. In the first edition, a desire to render every thing perfectly intelligible led the author to omit the use of technical terms as far as possible. We have no partiality ourselves for scientific treatises, overloaded with these ornamental appendages; yet it is very obvious, that as long as facts exist, those facts must have a name; as long as propositions of different forms are to be treated of, it will be very convenient, at least, to have distinguishing terms, which, when their meaning is once settled accurately by definitions, may ever after be employed, in a manner analogous to algebraic signs, instead of the definitions; and if these terms are etymologically significant of their scientific import, so much the better. This defect of the first edition has been corrected in the second, which is, in several other points, a decided improvement upon its predecessor.

It is of great importance, in an elementary work, to bring the subject treated of within as narrow limits as accuracy and perspicuity will admit. This condition has been fulfilled by Mr Walker. As to the scientific strictness of some of his means, we will not now decide, but reserve our remarks for particular instances. We have said that his arrangement differs from M. Legendre's. It differs in several particulars besides what we have already mentioned. For instance, the properties of the circle, of the triangle, of the polygon, &c. are treated of in connexion. The definitions, instead of being given in a body, occur as the nature of the subject demands them. The definitions themselves are given, in some in

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